3.21.20 \(\int e^{-e-x} (e^{7+x}+e^{e^{-x} x} (e^{2+x}+e^e (1-x)+e^2 (x-x^2))) \, dx\) [2020]

Optimal. Leaf size=25 \[ \left (e^5+e^{e^{-x} x}\right ) \left (1+e^{2-e} x\right ) \]

[Out]

(1+exp(1)^2/exp(exp(1))*x)*(exp(x/exp(x))+exp(5))

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Rubi [F]
time = 0.81, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int e^{-e-x} \left (e^{7+x}+e^{e^{-x} x} \left (e^{2+x}+e^e (1-x)+e^2 \left (x-x^2\right )\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[E^(-E - x)*(E^(7 + x) + E^(x/E^x)*(E^(2 + x) + E^E*(1 - x) + E^2*(x - x^2))),x]

[Out]

E^(7 - E)*x + Defer[Int][E^(-(((-1 + E^x)*x)/E^x)), x] + Defer[Int][E^(2*(1 - E/2) + x/E^x), x] - Defer[Int][x
/E^(((-1 + E^x)*x)/E^x), x] + Defer[Int][E^(2*(1 - E/2) - x + x/E^x)*x, x] - Defer[Int][E^(2*(1 - E/2) - x + x
/E^x)*x^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (e^{7-e}+e^{-e-x+e^{-x} x} \left (e^e+e^{2+x}+e^2 \left (1-e^{-2+e}\right ) x-e^2 x^2\right )\right ) \, dx\\ &=e^{7-e} x+\int e^{-e-x+e^{-x} x} \left (e^e+e^{2+x}+e^2 \left (1-e^{-2+e}\right ) x-e^2 x^2\right ) \, dx\\ &=e^{7-e} x+\int e^{-e-x+e^{-x} x} \left (e^{2+x}-e^e (-1+x)-e^2 (-1+x) x\right ) \, dx\\ &=e^{7-e} x+\int \left (e^{2-e+e^{-x} x}-e^{-x+e^{-x} x} (-1+x)-e^{2-e-x+e^{-x} x} (-1+x) x\right ) \, dx\\ &=e^{7-e} x+\int e^{2-e+e^{-x} x} \, dx-\int e^{-x+e^{-x} x} (-1+x) \, dx-\int e^{2-e-x+e^{-x} x} (-1+x) x \, dx\\ &=e^{7-e} x+\int e^{2 \left (1-\frac {e}{2}\right )+e^{-x} x} \, dx-\int e^{-e^{-x} \left (-1+e^x\right ) x} (-1+x) \, dx-\int e^{2 \left (1-\frac {e}{2}\right )-x+e^{-x} x} (-1+x) x \, dx\\ &=e^{7-e} x+\int e^{2 \left (1-\frac {e}{2}\right )+e^{-x} x} \, dx-\int \left (-e^{-e^{-x} \left (-1+e^x\right ) x}+e^{-e^{-x} \left (-1+e^x\right ) x} x\right ) \, dx-\int \left (-e^{2 \left (1-\frac {e}{2}\right )-x+e^{-x} x} x+e^{2 \left (1-\frac {e}{2}\right )-x+e^{-x} x} x^2\right ) \, dx\\ &=e^{7-e} x+\int e^{-e^{-x} \left (-1+e^x\right ) x} \, dx+\int e^{2 \left (1-\frac {e}{2}\right )+e^{-x} x} \, dx-\int e^{-e^{-x} \left (-1+e^x\right ) x} x \, dx+\int e^{2 \left (1-\frac {e}{2}\right )-x+e^{-x} x} x \, dx-\int e^{2 \left (1-\frac {e}{2}\right )-x+e^{-x} x} x^2 \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.12, size = 31, normalized size = 1.24 \begin {gather*} e^{7-e} x+e^{e^{-x} x} \left (1+e^{2-e} x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(-E - x)*(E^(7 + x) + E^(x/E^x)*(E^(2 + x) + E^E*(1 - x) + E^2*(x - x^2))),x]

[Out]

E^(7 - E)*x + E^(x/E^x)*(1 + E^(2 - E)*x)

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Maple [A]
time = 0.14, size = 32, normalized size = 1.28

method	result	size
risch	\(x \,{\mathrm e}^{-{\mathrm e}+7}+\left ({\mathrm e}^{2} x +{\mathrm e}^{{\mathrm e}}\right ) {\mathrm e}^{-{\mathrm e}+x \,{\mathrm e}^{-x}}\)	\(32\)
norman	\(\left ({\mathrm e}^{x} {\mathrm e}^{x \,{\mathrm e}^{-x}}+{\mathrm e}^{2} {\mathrm e}^{-{\mathrm e}} x \,{\mathrm e}^{x} {\mathrm e}^{x \,{\mathrm e}^{-x}}+{\mathrm e}^{-{\mathrm e}} {\mathrm e}^{2} {\mathrm e}^{5} x \,{\mathrm e}^{x}\right ) {\mathrm e}^{-x}\)	\(52\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((1-x)*exp(exp(1))+exp(1)^2*exp(x)+(-x^2+x)*exp(1)^2)*exp(x/exp(x))+exp(1)^2*exp(5)*exp(x))/exp(x)/exp(ex
p(1)),x,method=_RETURNVERBOSE)

[Out]

x*exp(-exp(1)+7)+(exp(2)*x+exp(exp(1)))*exp(-exp(1)+x*exp(-x))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((1-x)*exp(exp(1))+exp(1)^2*exp(x)+(-x^2+x)*exp(1)^2)*exp(x/exp(x))+exp(1)^2*exp(5)*exp(x))/exp(x)/
exp(exp(1)),x, algorithm="maxima")

[Out]

x*e^(-e + 7) + integrate(-(x^2*e^2 - x*(e^2 - e^e) - e^(x + 2) - e^e)*e^(x*e^(-x) - x - e), x)

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Fricas [A]
time = 0.36, size = 27, normalized size = 1.08 \begin {gather*} {\left (x e^{7} + {\left (x e^{2} + e^{e}\right )} e^{\left (x e^{\left (-x\right )}\right )}\right )} e^{\left (-e\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((1-x)*exp(exp(1))+exp(1)^2*exp(x)+(-x^2+x)*exp(1)^2)*exp(x/exp(x))+exp(1)^2*exp(5)*exp(x))/exp(x)/
exp(exp(1)),x, algorithm="fricas")

[Out]

(x*e^7 + (x*e^2 + e^e)*e^(x*e^(-x)))*e^(-e)

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Sympy [A]
time = 0.11, size = 31, normalized size = 1.24 \begin {gather*} \frac {x e^{7}}{e^{e}} + \frac {\left (x e^{2} + e^{e}\right ) e^{x e^{- x}}}{e^{e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((1-x)*exp(exp(1))+exp(1)**2*exp(x)+(-x**2+x)*exp(1)**2)*exp(x/exp(x))+exp(1)**2*exp(5)*exp(x))/exp
(x)/exp(exp(1)),x)

[Out]

x*exp(7)*exp(-E) + (x*exp(2) + exp(E))*exp(-E)*exp(x*exp(-x))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (22) = 44\).
time = 0.40, size = 45, normalized size = 1.80 \begin {gather*} {\left (x e^{\left (x e^{\left (-x\right )} - x - e + 2\right )} + x e^{\left (-x - e + 7\right )} + e^{\left (x e^{\left (-x\right )} - x\right )}\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((1-x)*exp(exp(1))+exp(1)^2*exp(x)+(-x^2+x)*exp(1)^2)*exp(x/exp(x))+exp(1)^2*exp(5)*exp(x))/exp(x)/
exp(exp(1)),x, algorithm="giac")

[Out]

(x*e^(x*e^(-x) - x - e + 2) + x*e^(-x - e + 7) + e^(x*e^(-x) - x))*e^x

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Mupad [B]
time = 1.43, size = 31, normalized size = 1.24 \begin {gather*} x\,{\mathrm {e}}^{7-\mathrm {e}}+{\mathrm {e}}^{x\,{\mathrm {e}}^{-x}-\mathrm {e}}\,\left ({\mathrm {e}}^{\mathrm {e}}+x\,{\mathrm {e}}^2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-exp(1))*exp(-x)*(exp(7)*exp(x) + exp(x*exp(-x))*(exp(2)*exp(x) - exp(exp(1))*(x - 1) + exp(2)*(x - x^
2))),x)

[Out]

x*exp(7 - exp(1)) + exp(x*exp(-x) - exp(1))*(exp(exp(1)) + x*exp(2))

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